Re: an axiom is...?

On Feb 28, 12:31�pm, Bill Dubuque <w...@xxxxxxxxxxxxxxxxxxxx> wrote:
Mensanator <mensana...@xxxxxxx> wrote:

I'm reading Lagarias' site on the 3x+1 problem:
http://www.cecm.sfu.ca/organics/papers/lagarias/paper/html/node2.html

I can't seem to find what he means by "trivial". Actually, I know
exactly what he means, I'm looking for where he defines what he means.

There's this:
<quote>
We call the sequence of iterates �the trajectory
of n. �There are three possible behaviors for
such trajectories when n > 0.

(i). �Convergent trajectory. Some T**(k)(n) = 1.

(ii). Non-trivial cyclic trajectory. The sequence
� � � T**(k)(n) eventually becomes periodic and
� � � T**(k)(n) != 1 for any k >= 1.

(iii). Divergent trajectory. lim_k->oo T**(k)(n) = oo.
</quote>

The (i),(ii),(iii) are definitions? Can I infer
from (i) and (ii) that the trivial cyclic trajectory
is what the Convergent trajectory converges to?

Yes, they are implicit definitions of the various classes
of named trajectories.

Ok.


Here's my actual question: what is the nature of the 1 in
T**(k)(n) = 1? Is it simply a term? Why isn't it some other
number like 2 or 3? Is the fact that it's 1 an axiom?

By the Pigeonhole Principle, any non-divergent trajectory
is eventually periodic ("cyclic"),

Ok.

so either it eventually
reaches the obvious ("trivial") cycle <1 2> or else it reaches
some other "non-trivial" cycle.

How was it obvious that <1 2> is trivial?
Empirically?

This elementary case analysis
is the genesis of the term "1" in that context.

Are you saying it is stated as 1 because the cycle is <1 2>?


The reason I ask is that 1 is NOT the trivial cyclic trajectory
when 3x+1 is extended to 3x+C. It is a theorem of Sequence Vector
Theory that the trivial cyclic trajectory is C for 3x+C. Am I
correct in saying that the 1 in T**(k)(n) = 1 is a derivation,
not a given?

You question makes no sense without a rigorous definiton of "given"
vs. "derivation". The meaning of Lagarias terms are completely
independent of any generalized problem or of any particular theory
that you may have developed. They are merely red herrings here.

So I can define "trivial" as something else when I talk about
3n+C?


--Bill Dubuque

[1] K. R> Matthews. The generalized 3x + 1 mapping.http://www.numbertheory.org/pdfs/survey.pdf
.

Relevant Pages

  • Re: an axiom is...?
    ... We call the sequence of iterates the trajectory ... It's a name or representative for that cycle. ... It is a theorem of Sequence Vector ... Theory that the trivial cyclic trajectory is C for 3x+C. ...
    (sci.math)
  • Re: an axiom is...?
    ... We call the sequence of iterates �the trajectory ... It's a cycle by inspection ... It is a theorem of Sequence Vector ... Theory that the trivial cyclic trajectory is C for 3x+C. ...
    (sci.math)
  • Re: an axiom is...?
    ... I'm reading Lagarias' site on the 3x+1 problem: ... We call the sequence of iterates the trajectory ... Non-trivial cyclic trajectory. ... Theory that the trivial cyclic trajectory is C for 3x+C. ...
    (sci.math)
  • Re: Interpolation of a discrete 3D trajectory
    ... I'm looking for a way to smooth out the edges of a 3D trajectory that is ... really just a sequence of points in space. ... Is there a reasonably easy way from, say, a 10 co-ordinate input to get ...
    (comp.lang.python)
  • Interpolation of a discrete 3D trajectory
    ... I'm looking for a way to smooth out the edges of a 3D trajectory that is ... really just a sequence of points in space. ... Is there a reasonably easy way from, say, a 10 co-ordinate input to get ...
    (comp.lang.python)